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Sagot :
To find the electromagnetic force between the two particles, we'll use Coulomb's law, which is given by the formula:
[tex]\[ F_e = \frac{k q_1 q_2}{r^2} \][/tex]
where:
- [tex]\(k\)[/tex] is Coulomb's constant, [tex]\(9.00 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2\)[/tex],
- [tex]\(q_1\)[/tex] and [tex]\(q_2\)[/tex] are the charges of the particles,
- [tex]\(r\)[/tex] is the separation distance between the charges.
Given data:
- [tex]\( q_1 = -1.87 \times 10^{-9} \, \text{C} \)[/tex]
- [tex]\( q_2 = -1.10 \times 10^{-9} \, \text{C} \)[/tex]
- [tex]\( r = 0.05 \, \text{m} \)[/tex]
Now substitute these values into the formula:
[tex]\[ F_e = \frac{(9.00 \times 10^9) \times (-1.87 \times 10^{-9}) \times (-1.10 \times 10^{-9})}{(0.05)^2} \][/tex]
First, calculate the numerator:
[tex]\[ (9.00 \times 10^9) \times (-1.87 \times 10^{-9}) \times (-1.10 \times 10^{-9}) \][/tex]
The product of the charges ([tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex]) will be positive because multiplying two negative numbers yields a positive result. Simplifying this, we get:
[tex]\[ (9.00 \times 10^9) \times (1.87 \times 10^{-9}) \times (1.10 \times 10^{-9}) = 9.00 \times 1.87 \times 1.10 \times 10^9 \times 10^{-9} \times 10^{-9} \][/tex]
[tex]\[ = 9.00 \times 1.87 \times 1.10 \times 10^{-9} \][/tex]
Next, calculate the denominator:
[tex]\[ (0.05)^2 = 0.0025 \][/tex]
Now substitute these into the expression for [tex]\( F_e \)[/tex]:
[tex]\[ F_e = \frac{9.00 \times 1.87 \times 1.10 \times 10^{-9}}{0.0025} \][/tex]
Calculate the values:
[tex]\[ 9.00 \times 1.87 = 16.83 \][/tex]
[tex]\[ 16.83 \times 1.10 = 18.513 \][/tex]
[tex]\[ F_e = \frac{18.513 \times 10^{-9}}{0.0025} \][/tex]
[tex]\[ F_e = 18.513 \times 10^{-9} \times \frac{1}{0.0025} \][/tex]
[tex]\[ F_e = 18.513 \times 10^{-9} \times 400 \][/tex]
[tex]\[ F_e = 7.4052 \times 10^{-6} \, \text{N} \][/tex]
Therefore, the electromagnetic force between the two particles is:
[tex]\[ \boxed{7.41 \times 10^{-6} \, \text{N}} \][/tex]
So the correct answer is:
C) [tex]\(7.41 \times 10^{-6} \, \text{N}\)[/tex]
[tex]\[ F_e = \frac{k q_1 q_2}{r^2} \][/tex]
where:
- [tex]\(k\)[/tex] is Coulomb's constant, [tex]\(9.00 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2\)[/tex],
- [tex]\(q_1\)[/tex] and [tex]\(q_2\)[/tex] are the charges of the particles,
- [tex]\(r\)[/tex] is the separation distance between the charges.
Given data:
- [tex]\( q_1 = -1.87 \times 10^{-9} \, \text{C} \)[/tex]
- [tex]\( q_2 = -1.10 \times 10^{-9} \, \text{C} \)[/tex]
- [tex]\( r = 0.05 \, \text{m} \)[/tex]
Now substitute these values into the formula:
[tex]\[ F_e = \frac{(9.00 \times 10^9) \times (-1.87 \times 10^{-9}) \times (-1.10 \times 10^{-9})}{(0.05)^2} \][/tex]
First, calculate the numerator:
[tex]\[ (9.00 \times 10^9) \times (-1.87 \times 10^{-9}) \times (-1.10 \times 10^{-9}) \][/tex]
The product of the charges ([tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex]) will be positive because multiplying two negative numbers yields a positive result. Simplifying this, we get:
[tex]\[ (9.00 \times 10^9) \times (1.87 \times 10^{-9}) \times (1.10 \times 10^{-9}) = 9.00 \times 1.87 \times 1.10 \times 10^9 \times 10^{-9} \times 10^{-9} \][/tex]
[tex]\[ = 9.00 \times 1.87 \times 1.10 \times 10^{-9} \][/tex]
Next, calculate the denominator:
[tex]\[ (0.05)^2 = 0.0025 \][/tex]
Now substitute these into the expression for [tex]\( F_e \)[/tex]:
[tex]\[ F_e = \frac{9.00 \times 1.87 \times 1.10 \times 10^{-9}}{0.0025} \][/tex]
Calculate the values:
[tex]\[ 9.00 \times 1.87 = 16.83 \][/tex]
[tex]\[ 16.83 \times 1.10 = 18.513 \][/tex]
[tex]\[ F_e = \frac{18.513 \times 10^{-9}}{0.0025} \][/tex]
[tex]\[ F_e = 18.513 \times 10^{-9} \times \frac{1}{0.0025} \][/tex]
[tex]\[ F_e = 18.513 \times 10^{-9} \times 400 \][/tex]
[tex]\[ F_e = 7.4052 \times 10^{-6} \, \text{N} \][/tex]
Therefore, the electromagnetic force between the two particles is:
[tex]\[ \boxed{7.41 \times 10^{-6} \, \text{N}} \][/tex]
So the correct answer is:
C) [tex]\(7.41 \times 10^{-6} \, \text{N}\)[/tex]
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